Hi,
so here's the code:
b = {0, -10*y}
NDSolve[ {[Rho][x, y,
t](D[ux[x, y, t], t] + ux[x, y, t]D[ux[x, y, t], x] +
uy[x, y, t]D[ux[x, y, t], y]) == -D[p[x, y, t], x] + [Rho][x,
y, t]b[[1]],
[Rho][x, y,
t](D[uy[x, y, t], t] + ux[x, y, t]D[uy[x, y, t], x] +
uy[x, y, t]D[uy[x, y, t], y]) == -D[p[x, y, t], y] + [Rho][x,
y, t]b[[2]],
D[[Rho][x, y, t], t] + ux[x, y, t]*D[[Rho][x, y, t], x] +
uy[x, y, t]*D[[Rho][x, y, t], y] == 0,
D[ux[x, y, t], x] + D[uy[x, y, t], y] == 0,
ux[0, y, t] == 0,
ux[10, y, t] == 0,
uy[x, 0, t] == 0,
uy[x, 10, t] == 0,
[Rho][x, y, 0] == 1,
p[x, 10, t] == 10}, {ux, uy, [Rho], p}, {x, 0, 10}, {y, 0,
10}, {t, 0, 10}]H
Boundary conditions translate to n dot u == 0 at the boundary. I wish I had a better way to post this code. I hope it's readable. [Rho] is the density, p is the pressure, u is the velocity and b is the body force, I just added gravity. The problem happens even in one dimension when you multiply de density times the velocity field.
Anyway as you sad, they are hard equations to solve. What's the procedure people usually do to solve them?
Thanks,
E.
P.S.: I'm attaching the notebook hoping it helps to read the stuff.
Attachments: